The Lambert function will be needed for this. On the other hand, your computer will very likely use numerical methods for evaluating the logarithm and the Lambert function anyway, so what's the point?
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Guess who it is.Apr 29 '12 at 17:49

Great! Since m and n are relatively large integers, this can be approximated as: $$ \frac{-m(m+n)}{n} W(\frac{-n}{e(m+n)})$$ Is there a simpler approximation to W, in this specific case?
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Erel Segal-HaleviApr 29 '12 at 18:23

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@ErelSegalHalevi - If $m<<n$, you can use $W(-1/e)=1$.
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nbubisApr 29 '12 at 18:35

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"Is there a simpler approximation to $W$?" - sure, you can take $$W(x)\approx \frac{ex}{1+\left(\frac1{e-1}-\frac1{\sqrt 2}+\frac1{\sqrt{2ex+2}}\right)^{-1}}$$ for instance. (The approximation is due to Serge Winitzki.)
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Guess who it is.Apr 30 '12 at 0:33